Skip to main content
SpringerLink
Log in

Dirac Equation with Position-Dependent Mass and Coulomb-like Field in Hausdorff Dimension

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

Dirac equation with spatially or position-dependent mass and an attractive Coulomb-like field is constructed in Hausdorff dimension of order \(0<\alpha \le 1\). The lower and upper components of the spinor wave function were derived in addition to the corresponding energy eigenvalues of the resulting relativistic equation. It was observed that, in Hausdorff radial dimension, the ground state energy of particles mass with spin half is enhanced which leads to an enhancement of electrons mass in agreement with recent theoretical observations and experimental data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructure (Les Editions de Physique, Les Ulis, 1988)

    Google Scholar 

  2. D.L. Smith, C. Mailhiot, Theory of semiconductor superlattice electronic structure. Rev. Mod. Phys. 62, 173–234 (1990)

    ADS  Google Scholar 

  3. G.T. Einevoll, Operator ordering in effective mass theory for heterostructures II. Strained systems. Phys. Rev. B 42, 3497 (1990)

    ADS  Google Scholar 

  4. P. Harrison, Quantum Wells, Wires and Dots (Wiley, New York, 2000)

    Google Scholar 

  5. R.A. El-Nabulsi, Some implications of position-dependent mass quantum fractional Hamiltonian in quantum mechanics. Eur. Phys. J. Plus 134, 192 (2019)

    Google Scholar 

  6. R.A. El-Nabulsi, Time-fractional Schrodinger equation from path integral and its implications in quantum dots and semiconductors. Eur. Phys. J. Plus 133, 394 (2018)

    Google Scholar 

  7. R.A. El-Nabulsi, Emergence of quasiperiodic quantum wave functions in Hausdorff dimensional crystals and improved intrinsic Carrier concentrations. J. Phys. Chem. Solids 127, 224–230 (2019)

    ADS  Google Scholar 

  8. F. Arias de Saavedra, J. Boronat, A. Polls, A. Fabrocini, Effective mass of one \(^{4}\text{ He }\) atom in liquid \(^{3}\text{ He }\). Phys. Rev. B 50, 4248 (1994)

    ADS  Google Scholar 

  9. W. Klopper, C. Samson, G. Tarczay, A. Csaszar, Equilibrium inversion barrier of NH\(_{\rm 3}\) from extrapolated coupled-cluster pair energies. J. Comput. Chem. 22, 1306 (2001)

    Google Scholar 

  10. J. Gordon, H. Zeiger, C. Townes, Molecular microwave oscillator and new hyperfine structure in the microwave spectrum of NH\(_{3}\). Phys. Rev. 95, 282 (1954)

    ADS  Google Scholar 

  11. J. Förster, A. Saenz, U. Wolff, Matrix algorithm for solving Schrödinger equations with position-dependent mass or complex optical potentials. Phys. Rev. E 86, 016701 (2012)

    ADS  Google Scholar 

  12. F.Q. Zhao, X.X. Liang, S.L. Ban, Influence of the spatially dependent effective mass on bound polarons in finite parabolic quantum wells. Eur. Phys. J. B 33, 3–8 (2003)

    ADS  Google Scholar 

  13. S. Meyur, S. Maji, S. Debnath, Analytical solution of the Schrödinger equation with spatially varying effective mass for generalized Hylleraas potential. Adv. High Energy Phys. 2014, Article ID 952597 (7 pages) (2014)

  14. A.R. Plastino, A. Rigo, M. Casas, F. Garcias, A. Plastino, Supersymmetric approach to quantum systems with position-dependent effective mass. Phys. Rev. A 60, 4318 (1999)

    ADS  Google Scholar 

  15. B. Gonul, B. Gonul, D. Tutco, O. Ozer, Supersymmetric approach to exactly solvable systems with position-dependent effective mass. Mod. Phys. Lett. A 17, 2057 (2002)

    ADS  MATH  Google Scholar 

  16. S.-H. Dong, M. Lozada-Cassou, Exact solutions of the Schrödinger equation with the position-dependent mass for a hard-core potential. Phys. Lett. A 337, 313–320 (2005)

    ADS  MATH  Google Scholar 

  17. S.-H. Dong, J.J. Pena, C. Pacheco-Garcia, J. Garcia-Ravelo, Algebraic approach to the position-dependent mass Schrodinger equation for a singular oscillator. Mod. Phys. Lett. A 22, 1039 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  18. G.H. Sun, D. Popov, O. Camacho-Nieto, S.H. Dong, Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well. Chin. Phys. B 24, 100303 (2015)

    ADS  Google Scholar 

  19. B.J. Falaye, F.A. Serrano, S.-H. Dong, Fisher information for the position-dependent mass Schrödinger system. Phys. Lett. A 380, 267–271 (2016)

    ADS  MATH  Google Scholar 

  20. J. Yu, S.-H. Dong, Exactly solvable potentials for the Schrodinger equation with spatially dependent mass. Phys. Lett. A 325, 194–198 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  21. S.C. Cruz, O. Rosas-Ortiz, Position-dependent mass oscillators and coherent states. J. Phys. A Math. Theor. 42, 185205 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  22. R.N. Costa-Filho, M.P. Almeida, G.A. Farias, J.S. Andrade Jr., Displacement operator for quantum systems with position-dependent mass. Phys. Rev. A 84, 050102 (2011)

    ADS  Google Scholar 

  23. J.J. Peña, J. Morales, J. Garcia-Ravelo, L. Arcos-Diaz, Schrödinger equation with position-dependent mass: staggered mass distributions. Int. J. Phys. Math. Sci. 11, 324–327 (2017)

    Google Scholar 

  24. X.-Y. Gu, S.-H. Dong, Z.-Q. Ma, Energy spectra for modified Rosen–Morse potential solved by the exact quantization rule. J. Phys. A Math. Theor. 42, 035303 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  25. S.H. Dong, X.Y. Gu, Arbitrary \(l\) state solutions of the Schrödinger equation with the Deng–Fan molecular potential. J. Phys. Conf. Ser. 96, 012109 (2008)

    Google Scholar 

  26. S.H. Dong, G.H. Sun, D. Popov, Group theory approach to the Dirac equation with a Coulomb plus scalar potential in D\(+\)1 dimensions. J. Math. Phys. 44, 4467 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  27. R. Renan, M.H. Pacheco, C.A.S. Almeida, Treating some solid problems with the Dirac equation. J. Phys. A Math. Theor. 33, L509 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  28. O. von Roos, Position-dependent effective masses in semiconductor theory. Phys. Rev. B 27, 7547 (1983)

    ADS  Google Scholar 

  29. M.K. Bahar, F. Yasuk, Bound states of the Dirac equation with position-dependent mass for the Eckart potential. Chin. Phys. B 22, 010301 (2013)

    ADS  Google Scholar 

  30. O. Panella, S. Biondini, A. Arda, New exact solution of the one dimensional Dirac Equation for the Woods–Saxon potential within the effective mass case. J. Phys. A Math. Theor. 43, 325302 (2010)

    MathSciNet  MATH  Google Scholar 

  31. S. Alpdogan, A. Havare, Dirac particle for the position dependent mass in the generalized asymmetric Woods–Saxon potential. Adv. High Energy Phys. 2014, Article ID 973847, 10 pages (2014)

  32. O. Aydogdu, A. Arda, R. Sever, Effective-mass Dirac equation for Woods–Saxon potential: scattering, bound states, and resonances. J. Math. Phys. 53, 042106 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  33. O. Aydogdu, A. Arda, R. Sever, Scattering of a spinless particle by an asymmetric Hulthén potential within the effective mass formalism. J. Math. Phys. 53, 102111 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  34. X.-L. Peng, J.-Y. Liu, C.-S. Jia, Approximation solution of the Dirac equation with position-dependent mass for the generalized Hulthén potential. Phys. Lett. A 352, 478–483 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  35. A. Arda, R. Sever, C. Tezcan, Approximate analytical solutions of the effective mass Dirac equation for the generalized Hulthén potential with any \(\kappa \)-value. Cent. Eur. J. Phys. 8, 843–849 (2010)

    Google Scholar 

  36. B. Mandelbrot, Fractal Geometry of Nature (Henry Holt & Co., New York, 1982)

    MATH  Google Scholar 

  37. G. Calcagni, G. Nardelli, M. Scalisi, Quantum mechanics in fractional and other anomalous spacetimes. J. Math. Phys. 53, 102110–102125 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  38. G. Losa, T. Nonnenmacher, Self-similarity and fractal irregularity in pathologic tissues. Mod. Pathol. 9, 174–182 (1996)

    Google Scholar 

  39. R.A. El-Nabulsi, Modifications at large distances from fractional and fractal arguments. Fractals 18, 1 (2010)

    MathSciNet  MATH  Google Scholar 

  40. V.E. Tarasov, Continuous medium model for fractal media. Phys. Lett. A 336, 167–174 (2005)

    ADS  MATH  Google Scholar 

  41. M. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Dover Publications, Mineola, 1991)

    MATH  Google Scholar 

  42. R.A. Blaya, J.B. Reyes, On boundary value problems for perturbed Hermitian matrix Dirac equations in a fractal domain. Bull. Belg. Math. Soc. Simon Stevin 21, 733–746 (2014)

    MathSciNet  MATH  Google Scholar 

  43. R.A. Blaya, J.B. Reyes, F. Brackx, H. De Schepper, F. Sommen, A Hilbert transform for matrix functions on fractal domains. Complex Anal. Oper. Theor. 6, 359–372 (2012)

    MathSciNet  MATH  Google Scholar 

  44. R.A. Blaya, J.B. Reyes, F. Brackx, H. De Schepper, F. Sommen, Boundary value problems associated to a Hermitian Helmholtz equation. J. Math. Anal. Appl. 389, 1268–1279 (2012)

    MathSciNet  MATH  Google Scholar 

  45. I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)

    MATH  Google Scholar 

  46. K.S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equation (Wiley, New York, 1993)

    MATH  Google Scholar 

  47. N. Laskin, Fractional Schrodinger equation. Phys. Rev. E 66, 056108 (2002)

    ADS  MathSciNet  Google Scholar 

  48. N. Laskin, Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 268, 298–305 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  49. M.M.I. Nayga, J.P.H. Esguerra, Levy path integral approach to the fractional Schrödinger equation with delta-perturbed infinite square well. Int. J. Mod. Phys. Conf. Ser. 36, 1560015–1560020 (2015)

    MATH  Google Scholar 

  50. Y. Zhang, X. Liu, M.R. Belic, W. Zhong, Y. Zhang, M. Xiao, Propagation dynamics of a light beam in a fractional Schrödinger equation. Phys. Rev. Lett. 115, 180403–180407 (2015)

    ADS  Google Scholar 

  51. S. Longhi, Fractional Schrödinger equation in optics. Opt. Lett. 40, 1117–1120 (2015)

    ADS  Google Scholar 

  52. A. Liemert, A. Kienle, Fractional Schrödinger equation in the presence of the linear potential. Mathematics 4, 31–45 (2016)

    MATH  Google Scholar 

  53. Y. Luchko, Fractional Schrödinger equation for a particle moving in a potential well. J. Math. Phys. 54, 012111 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  54. R.A. El-Nabulsi, Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal. J. Stat. Phys. 172, 1617–1640 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  55. R.A. El-Nabulsi, On the fractional minimal length Heisenberg-Weyl uncertainty relation from fractional Riccati generalized momentum operator. Chaos Solitons Fract. 42, 84–88 (2009)

    ADS  MATH  Google Scholar 

  56. R.A. El-Nabulsi, Complexified quantum field theory and mass without mass from multidimensional fractional actionlike variational approach with time-dependent fractional exponent. Chaos Solitons Fract. 42, 2384–2398 (2009)

    ADS  Google Scholar 

  57. R.A. El-Nabulsi, Fractional quantum Euler–Cauchy equation in the Schrödinger picture, complexified harmonic oscillators and emergence of complexified Lagrangian and Hamiltonian dynamics. Mod. Phys. Lett. B 23, 3368 (2009)

    ADS  MATH  Google Scholar 

  58. R.A. El-Nabulsi, The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars. Appl. Math. Comput. 218, 2837–2849 (2011)

    MathSciNet  MATH  Google Scholar 

  59. R.A. El-Nabulsi, G.-C. Wu, Fractional complexified field theory from Saxena–Kumbhat fraction integral, fractional derivative of order Alfa Beta and dynamical fractional integral exponent. Afr. Diaspora J. Math. 13, 45–61 (2012)

    MathSciNet  MATH  Google Scholar 

  60. R.A. El-Nabulsi, Gravitons in fractional action cosmology. Int. J. Theor. Phys. 51, 3978 (2012)

    MathSciNet  MATH  Google Scholar 

  61. R.A. El-Nabulsi, Fractional derivatives generalization of Einstein’s field equations. Indian J. Phys. 87, 195–200 (2013)

    ADS  Google Scholar 

  62. O. Khan, N. Khan, Computable solution of fractional kinetic equations using Mathieu-type series. Adv. Differ. Equ. 2019, 234 (2019)

    MathSciNet  MATH  Google Scholar 

  63. A. Antagana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fract. 102, 395–406 (2017)

    ADS  MathSciNet  Google Scholar 

  64. F.B. Tatom, The relationship between fractional calculus and fractals. Fractals 03, 217 (1995)

    MathSciNet  MATH  Google Scholar 

  65. S. Butera, M. Di Paola, A physically based connection between fractional calculus and fractal geometry. Ann. Phys. 350, 146–158 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  66. A. Rocco, B.J. West, Fractional calculus and the evolution of fractal phenomena. Phys. A 265, 535–546 (1999)

    Google Scholar 

  67. A.K. Golmankhaneh, D. Baleanu, Fractal calculus involving Gauge function. Commun. Nonlinear Sci. 37, 125–130 (2016)

    MathSciNet  Google Scholar 

  68. A.K. Golmankhaneh, D. Baleanu, Non-local integrals and derivatives on fractal sets with applications. Open Phys. 14, 542–548 (2016)

    Google Scholar 

  69. A.K. Golmankhaneh, C. Tunc, On the Lipschitz condition in the fractal calculus. Chaos Solitons Fract. 95, 140–147 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  70. M. Zubair, M.J. Mughal, Q.A. Naqvi, On electromagnetic wave propagation in fractional space. Nonlinear Anal Real World Appl. 12, 2844–2850 (2011)

    MathSciNet  MATH  Google Scholar 

  71. F.H. Stillinger, Axiomatic basis for spaces with noninteger dimension. J. Math. Phys. 18, 1224–1234 (1977)

    ADS  MathSciNet  MATH  Google Scholar 

  72. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space. J. Electromagn. Waves Appl. 25, 1481–1491 (2011)

    Google Scholar 

  73. M. Zubair, M.J. Mughal, Q.A. Naqvi, Electromagnetic wave propagation in fractional space, Electromagnetic Fields and Waves in Fractional Dimensional Space, ed. by M. Zubair, M.J. Mughal, Q.A. Naqvi, Springer Briefs in Applied Sciences and Technology (Springer, Berlin, 2012)

    MATH  Google Scholar 

  74. Q.A. Naqvi, M. Zubair, On cylindrical model of electrostatic potential in fractional dimensional space. Optik Int. J. Light Electron Opt. 127, 3243–3247 (2016)

    Google Scholar 

  75. M. Zubair, M.J. Mughal, Q.A. Naqvi, The wave equation and general plane wave solutions in fractional space. Prog. Electromagn. Res. Lett. 19, 137–146 (2010)

    Google Scholar 

  76. M. Zubair, Y.S. Ang, L.K. Ang, Thickness dependence of space-charge-limited current in spatially disordered organic semiconductors. IEEE Trans. Electron Dev. 65(8), 3421–3429 (2018)

    ADS  Google Scholar 

  77. M. Zubair, Y.S. Ang, L.K. Ang, Fractional Fowler–Nordheim law for field emission from rough surface with nonparabolic energy dispersion. IEEE Trans. Electron Dev. 65(6), 2089–2095 (2018)

    ADS  Google Scholar 

  78. M. Zubair, L.K. Ang, Fractional dimensional Child–Langmuir law for a rough cathode. Phys. Plasmas 23(7), 072118 (2016)

    ADS  Google Scholar 

  79. P. Strange, Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics (Cambridge University Press, Cambridge, 1998)

    Google Scholar 

  80. W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 2nd edn. (Springer, Berlin, 1997)

    MATH  Google Scholar 

  81. R. Schafer, An Introduction to Nonassociative Algebras (Academic Press, New York, 1966)

    MATH  Google Scholar 

  82. J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics (McGraw Hill, New York, 1965)

    MATH  Google Scholar 

  83. A.D. Alhaidari, H. Bahlouli, A. Al-Hasan, Dirac and Klein–Gordon equations with equal scalar and vector potentials. Phys. Lett. A 349, 87–97 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  84. E. Blomquist, T. Boman, The Dirac equation for a particle in a spherical box potential with application in bag modeling, SA104X Degree Project in Engineering Physics, First Level Department of Theoretical Physics Royal Institute of Technology (KTH), 2015

  85. M. Eshghi, S.M. Ikhdair, Exact solutions of a spatially-dependent mass Dirac equation for Coulomb field plus tensor interaction via Laplace transformation method. Adv. High Energy Phys. 2012, Article ID 873619 (2012)

  86. S.M. Ikhdair, R. Sever, Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential. Appl. Math. Comput. 216, 545–555 (2010)

    MathSciNet  MATH  Google Scholar 

  87. M.K. Bahar, F. Yasuk, Exact solutions of the mass-dependent Klein–Gordon equation with the vector quark-antiquark interaction and harmonic oscillator potential. Adv. High Energy Phys. 2013, Article ID 814985 (2013)

  88. S.M. Ikhdair, Exact Klein–Gordon equation with spatially dependent masses for unequal scalar–vector Coulomb-like potentials. Eur. Phys. J. A 40, 143–149 (2009)

    ADS  Google Scholar 

  89. S. Pezzini, M.R. van Delft, L. Schoop, B. Lotsch, A. Carrington, M.I. Katsnelson, N.E. Hussey, S. Wiedmann, Unconventional mass enhancement around the Dirac nodal loop in ZrSiS. Nat. Phys. 14, 178–183 (2018)

    Google Scholar 

  90. J. Peng, X.M. Gu, G.T. Zhou, W. Wang, J.Y. Liu, T. Wang, Z.Q. Mao, X.S. Wu, S. Dong, Electron mass enhancement and magnetic phase separation near the Mott transition in double-layer ruthenates. Front. Phys. 13, 137108 (2018)

    ADS  Google Scholar 

  91. T. Matsuura, T. Hotta, Relation between electron mass enhancement and potential shape: numerical analysis of two-site anharmonic Holstein–Hubbard model. J. Phys. Conf. Ser. 592, 012144 (2015)

    Google Scholar 

  92. P.L. Hagelstein, Electron mass enhancement and the Widom–Larsen model. J. Condens. Matter Nucl. Sci. 12, 18–40 (2013)

    Google Scholar 

  93. T. Watanabe, J.B. Murphy, J. Rose, Y. Shen, T. Tsang, X.J. Wang, H.P. Freund, The first experimental observation of FEL amplifier efficient improvement using electron beam energy detuning at the NSLS SDL, in Proceedings of FEL (BESSY, Berlin, Germany, 2006), pp. 190–196

  94. Y.M. Zhu, D.W. Zhang, Y.C. Yang, S.L. Zhuang, K. Hirakawa, The effective mass of electron enhancement in \(\Gamma \)-Valley in bulk GaAs under very high electric field investigated by time-domain terahertz spectroscopy, in Proceedings of SPIE 7277, Photonics and Optoelectronics Meeting (POEM) 2008: Terahertz Science and Technology, 727220H, 16 February 2009, https://doi.org/10.1117/12.821094

  95. W. Wunderlich, H. Ohta, K. Koumoto, Enhanced effective mass in doped SrTiO\(_{\rm 3}\) and related perovskites. Phys. B Condens. Matter 404, 2202–2212 (2009)

    ADS  Google Scholar 

  96. V. Todorinov, P. Bosso, S. Das, Relativistic generalized uncertainty principle. Ann. Phys. 405, 92–100 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  97. A. Shokrollahi, Free motion of a Dirac particle with a minimum uncertainty in position. Rep. Math. Phys. 70, 1–13 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  98. R.A. El-Nabulsi, Some implications of three generalized uncertainty relations in statistical mechanics of an ideal gas. Eur. Phys. J. P135, 34 (2020)

    Google Scholar 

Download references

Acknowledgements

The author expresses his gratitude for the group of anonymous referees for their useful comments and valuable suggestions.

Author information

Authors and Affiliations

  1. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece

    Rami Ahmad El-Nabulsi

Authors
  1. Rami Ahmad El-Nabulsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rami Ahmad El-Nabulsi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Nabulsi, R.A. Dirac Equation with Position-Dependent Mass and Coulomb-like Field in Hausdorff Dimension. Few-Body Syst 61, 10 (2020). https://doi.org/10.1007/s00601-020-1544-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00601-020-1544-6

Search

Navigation